Galois groups of co-abelian ball quotient covers

Abstract

If $X^{\prime }=(\mathbb{B}/\Gamma {)}^{\prime }$ is a torsion free toroidal compactification of a discrete ball quotient $X_o=\mathbb{B}/\Gamma$ and $\xi :(X^{\prime },T=X^{\prime }\setminus X_o)\to (X,D=\xi (T))$ is the blow-down of the $(-1)$-curves to the corresponding minimal model, then $G^{\prime }=Aut(X^{\prime },T)$ coincides with the finite group $G=Aut(X,D)$. In particular, for an elliptic curve $E$ with endomorphism ring $R=End(E)$ and a split abelian surface $X=A=E\times E$, $G$ is a finite subgroup of $Aut(A)={\mathcal{T}}_A⋋GL(2,R)$, where $({\mathcal{T}}_A,+)\simeq (A,+)$ is the translation group of $A$ and $GL(2,R)=\{g\in R_{2\times 2}\Vert \det (g)\in R^{\ast }\}$. The present work classifies the finite subgroups $H$ of $Aut(A=E\times E)$ for an arbitrary elliptic curve $E$. By the means of the geometric invariants theory, it characterizes the Kodaira-Enriques types of $A/H\simeq (\mathbb{B}/\Gamma {)}^{\prime }/H$, in terms of the fixed point sets of $H$ on $A$. The abelian and the K3 surfaces $A/H$ are elaborated in \cite{KN}. The first section provides necessary and sufficient conditions for $A/H$ to be a hyper-elliptic, ruled with elliptic base, Enriques or a rational surface. In such a way, it depletes the Kodaira-Enriques classification of the finite Galois quotients $A/H$ of a split abelian surface $A=E\times E$. The second section derives a complete list of the conjugacy classes of the linear automorphisms $g\in GL(2,R)$ of $A$ of finite order, by the means of their eigenvalues. The third section classifies the finite subgroups $H$ of $GL(2,R)$. The last section provides explicit generators and relations for the finite subgroups $H$ of $Aut(A)$ with K3, hyper-elliptic, rules with elliptic base or Enriques quotients $A/H\simeq (\mathbb{B}/\Gamma {)}^{\prime }/H$.

References in corpus (1)

• Applications of principal isogenies to constructions of ball quotient surfaces

Cited by in corpus (2)

• Applications of principal isogenies to constructions of ball quotient surfaces
• Toroidal compactifications of torsion free local complex hyperbolic spaces